Mister-Meeseeks · Oct 16, 2026 · Oct 15, 2026
diff --git a/research/triadic-query-pair/CERTIFICATE.json b/research/triadic-query-pair/CERTIFICATE.json
@@ -0,0 +1,153 @@
+{
+  "base_commit": "0b1cccf86e2f5132ed286a26ef0a681a495740ea",
+  "baseline_invoice": {
+    "aligned_xor_scans": 4,
+    "child_volume_fraction_each": "1/3",
+    "child_width_fraction_each": "1/3",
+    "kernel": 3,
+    "ordered_affine_events": 24,
+    "ordinary_children": 4,
+    "persistent_private_rows": 0,
+    "query_powers": [
+      -3,
+      3,
+      3,
+      -3
+    ]
+  },
+  "claimed_scope": "three-query completed native DATA shear for a=\u00b13\u00b72^j with odd moduli coprime to 3; paid lowering, not a supplier",
+  "finite_arrays": {
+    "full_dirty_endpoint_columns": 5766,
+    "moduli": [
+      5,
+      7
+    ]
+  },
+  "fixture": {
+    "a": 3,
+    "j": 0,
+    "sign": 1
+  },
+  "format": "triadic-native-query-pair-certificate-v1",
+  "fused_invoice": {
+    "aligned_xor_scans": 2,
+    "child_volume_fraction_each": "1/3",
+    "child_width_fraction_each": "1/3",
+    "kernel": 3,
+    "ordered_affine_events": 18,
+    "ordinary_children": 3,
+    "persistent_private_rows": 0,
+    "query_powers": [
+      -3,
+      6,
+      -3
+    ]
+  },
+  "implementation_sha256": {
+    "PROOF.md": "7a86fa2db00b6b407141c27ee8547adf9a1068a21740812776d03b8961da0e0a",
+    "README.md": "4fb43ca8530a309f4a0d3b741c97e910a5bbcdb4183c9d11e11d2bde16aa5aca",
+    "independent.py": "4bd2d0e0601ae1917d95ca197c30b0d109b23b8c5a6144b34948cd44a1de4128",
+    "program.py": "3948038aec1c6f4d938d71b6aa2d6dd91b05146994c4740a35f67972f96d7990",
+    "verify.py": "4eb4f965c0b64be71b69740cf4f3e00a9995ec20490a1bed1220bb8138e2c260"
+  },
+  "kernel": 3,
+  "laurent": {
+    "baseline_invoice": {
+      "aligned_xor_scans": 4,
+      "child_volume_fraction_each": "1/3",
+      "child_width_fraction_each": "1/3",
+      "kernel": 3,
+      "ordered_affine_events": 24,
+      "ordinary_children": 4,
+      "persistent_private_rows": 0,
+      "query_powers": [
+        -3,
+        3,
+        3,
+        -3
+      ]
+    },
+    "centered_survivors": 0,
+    "fused_invoice": {
+      "aligned_xor_scans": 2,
+      "child_volume_fraction_each": "1/3",
+      "child_width_fraction_each": "1/3",
+      "kernel": 3,
+      "ordered_affine_events": 18,
+      "ordinary_children": 3,
+      "persistent_private_rows": 0,
+      "query_powers": [
+        -3,
+        6,
+        -3
+      ]
+    },
+    "scope": "finite corroboration; universal exclusion is the written proof",
+    "signed_symbolic_program_pairs": 16,
+    "two_query_products_checked": 91320
+  },
+  "lowerings": {
+    "admitted_moduli": [
+      5,
+      7,
+      11,
+      13,
+      17,
+      19,
+      23,
+      25,
+      35,
+      49,
+      55,
+      77,
+      121
+    ],
+    "denominator_odd_part": 3,
+    "excluded_moduli": [
+      3,
+      9,
+      15,
+      21,
+      27,
+      33,
+      45,
+      63,
+      75,
+      81,
+      99
+    ],
+    "kernel": 3,
+    "mixed_rational_vectors": 240,
+    "modular_columns": 1560,
+    "rational_columns": 120,
+    "signed_powers": [
+      -3,
+      3,
+      -6,
+      6,
+      -12,
+      12,
+      -24,
+      24,
+      -48,
+      48
+    ]
+  },
+  "negative_controls": [
+    "omit_fused_event_0",
+    "omit_fused_event_1",
+    "omit_fused_event_2",
+    "omit_fused_event_3",
+    "omit_fused_event_4",
+    "central_power_not_doubled",
+    "first_query_wrong_sign",
+    "kernel3_excludes_multiples_of_three",
+    "modulus_three_execution_rejected",
+    "even_modulus_rejected",
+    "first_difference_two_query_scope",
+    "aliased_rows_rejected",
+    "complements_pr840_non_dyadic_exclusion"
+  ],
+  "new_kappa_claimed": false,
+  "source_manifest_canonical_sha256": "abe5b034f4c17e022cc3f90a4743cf16e5415989a7cecfc9cae454f251b2ab05"
+}
diff --git a/research/triadic-query-pair/NOTICE.md b/research/triadic-query-pair/NOTICE.md
@@ -0,0 +1,15 @@
+# Notice — research/triadic-query-pair
+
+Prepared by Wentao Li (GitHub VictorLiwentao, X @Is__Victor) with substantial
+AI assistance. New material is Apache-2.0 under the repository LICENSE.
+
+Predecessor credit:
+
+- Wentao Li / PR840 (`research/dyadic-query-pair`) — three-query dyadic fusion,
+  sharp minimum, typed costs, and the explicit non-dyadic exclusion this package
+  completes for kernel 3.
+- Karl Appel / applecrusher / PR521 — native unipotent ADDRESS query lowering.
+- commdata2338 / PR677 — completed DATA-shear boundaries without a private row.
+- Retained OpenAI tape / stream framework as vendored and cited by PR840.
+
+No independent human review is claimed. No selected frontier or kappa change.
diff --git a/research/triadic-query-pair/PROOF.md b/research/triadic-query-pair/PROOF.md
@@ -0,0 +1,102 @@
+# Three queries for a centered odd-kernel native BIT shear
+
+This is a local recursive-interface construction and a scoped query lower
+bound. Ordinary ADDRESS interchange and ordered affine tape streams remain
+premises. There is no new supplier, kappa, or formal multiplication proof.
+
+It answers the explicit non-dyadic limitation left open by
+[PR840](https://github.com/CrocSwap/integer-mult-bounds/pull/840)
+(`research/dyadic-query-pair`): the PR840 lowering specializes to every odd
+modulus for dyadic powers, and states that a non-dyadic power introduces prime
+exclusions into that particular word. Here the odd kernel is fixed to **3**,
+the completed endpoint uses `a = ±3·2^j`, and every excluded modulus is named.
+
+## Object
+
+Use the same six ADDRESS fields and unipotent push `h` as PR521/PR840, with
+
+```text
+s=(1,0,1,-1,0,-1)^T, r=(1,0,-1,-1,0,1)^T,
+M(k)=I6+(k/4)s r^T,   h = array permutation of M(1).
+```
+
+For each integer `j >= 0` and sign `σ ∈ {±1}`, set `a = σ·3·2^j`. The completed
+DATA endpoint is
+
+```text
+F_a(x,y,z)=(x+(h^a+h^-a)y, y, z).                 (1)
+```
+
+VALUE addition is over F2. Source `y` and every spectator remain arbitrarily
+dirty. The target requires exclusive access until the completed boundary.
+
+## Lowering of every signed odd-kernel power
+
+Put `c = k/4` for `k = ±3·2^j`. The same seven-event word as PR840 applies:
+
+```text
+z2 <- z2-z0
+z3 <- z3+z0
+z5 <- c*z5+(1+c)*z0-c*z2-c*z3
+CHILD_C(0,5)
+z2 <- z2+z0
+z3 <- z3-z0
+z5 <- -(1/c)*z5+(1/c-1)*z0+z2+z3.                  (2)
+```
+
+The algebraic identity `M(k)` is unchanged for any nonzero integer `k`. The
+diagonal units are `1`, `c`, and `-1/c`. For `k = ±3·2^j` every numerator and
+denominator factor is a power of two times a power of three (up to sign).
+Therefore (2) specializes to **every odd modulus not divisible by 3**. Moduli
+divisible by 3, and every even modulus, are excluded from this particular
+lowering. No all-odd-radix claim is made.
+
+## Completed pair fusion
+
+The chronological fused program is identical in form to PR840:
+
+```text
+x <- h^-a x
+x ^= y
+x <- h^(2a) x
+x ^= y
+x <- h^-a x.                                      (3)
+```
+
+The three target permutations multiply to the identity, and the VALUE endpoint
+is (1). Relative to two separate completed PR677-style shears for `h^a` and
+`h^-a` (four queries, four XOR scans), the fused word uses **three ordinary
+children, eighteen affine events and two XOR scans**.
+
+## Three queries are necessary
+
+The PR840 determinant argument applies verbatim with any nonzero integer unit
+power `a` on an odd full-order orbit: zero or one query cannot match (1); two
+queries force opposite exponents and then collapse to the identity after
+comparing Laurent coefficients. The finite two-query census in the verifier
+corroborates the `a = ±3` case on dimensions 2–4; the written argument is the
+universal exclusion inside the stated query/XOR alphabet.
+
+## Complete typed costs
+
+With the same notation as PR840 for child cost `C`, lowering `L_k`, XOR `X`
+and routing `R`,
+
+```text
+B_a <= 4 C + 2 L_a + 2 L_-a + 4 X + R_b,
+F_a <= 3 C + 2 L_-a + L_2a + 2 X + R_f.
+```
+
+The middle lowering uses `k = 2a = ±3·2^{j+1}` and therefore a different
+coefficient table. No physical runtime saving is asserted for every finite
+implementation. No supplier admission, precision/tape transfer or new kappa
+follows.
+
+## Evidence boundary
+
+The checker compares (2) to the independent closed form `M(k)`, specializes to
+named admitted and excluded moduli, checks Laurent endpoints and a two-query
+census, and replays dirty BIT columns on `(Z/5)^6` and `(Z/7)^6`. An independent
+module rebuilds the matrix identity `M(-a)M(2a)M(-a)=I` without importing the
+primary verifier. Negative controls omit fused events, break the doubled middle
+power, and reject modulus 3 and even rings.
diff --git a/research/triadic-query-pair/README.md b/research/triadic-query-pair/README.md
@@ -0,0 +1,84 @@
+# Fuse a completed native BIT shear pair at odd kernel 3
+
+**Research construction / recursive interface; no new kappa.** Complements
+[PR840](https://github.com/CrocSwap/integer-mult-bounds/pull/840)
+(`research/dyadic-query-pair`), which fuses two completed DATA shears for
+dyadic powers `a=±2^j` with an all-odd-radix lowering and explicitly leaves
+non-dyadic powers to introduce prime exclusions. This package fixes the odd
+kernel to **3**, takes `a=±3·2^j`, and prices the same three-query fusion with
+every excluded modulus named.
+
+```text
+x <- h^-a x
+x ^= y
+x <- h^(2a) x
+x ^= y
+x <- h^-a x
+```
+
+| Complete cost item | Separate pair | Fused odd-kernel pair |
+|---|---:|---:|
+| Ordinary recursive children | 4 | 3 |
+| Ordered affine events | 24 | 18 |
+| Aligned XOR scans | 4 | 2 |
+| Extra full private rows | 0 | 0 |
+
+Each query lowers to six ordered affine events and one ordinary ADDRESS
+interchange child. Diagonal units involve the factor 3, so the lowering
+specializes to every **odd modulus not divisible by 3**. Multiples of 3 and
+even rings are rejected by the verifier. These are explicit interface counts,
+not measured runtime or a selected-supplier optimization.
+
+## Reproduce
+
+Python 3.12, standard library only, from the repository root:
+
+```sh
+python3.12 -I -B research/triadic-query-pair/verify.py
+python3.12 -I -O -B research/triadic-query-pair/verify.py
+python3.12 -I -B research/triadic-query-pair/independent.py
+python3.12 -I -B research/triadic-query-pair/test_package.py
+python3.12 -B scripts/research_package.py --package research/triadic-query-pair
+```
+
+Expected:
+
+```text
+PASS_TRIADIC_QUERY_PAIR children 4->3; affine 24->18; XOR 4->2; kernel=3 ... no_new_kappa
+PASS_INDEPENDENT_TRIADIC_QUERY_PAIR children=4->3 xor=4->2 kernel=3 ...
+PASS_TRIADIC_QUERY_PAIR_PACKAGE_TESTS ...
+PASS_REGISTERED_RESEARCH_PACKAGE research/triadic-query-pair outcome=completed
+```
+
+Default verify is read-only and compares a fresh rebuild to
+[CERTIFICATE.json](CERTIFICATE.json). Use `--write` only to regenerate that
+receipt. Admission uses `need()` (not `assert`); `-O` also passes.
+
+## Scope and overlap
+
+Base commit `0b1cccf86e2f5132ed286a26ef0a681a495740ea`. Direct predecessor:
+PR840 / `research/dyadic-query-pair` (Wentao Li), which retains PR521/PR677
+credit. This package does not reopen leave-alone tooling through
+[#2487](https://github.com/CrocSwap/integer-mult-bounds/pull/2487), the pending
+`mutable-source-789q-invoice` work, or ledger Deferred objects.
+
+Closest prior: PR840 proves the dyadic all-odd-radix case and the three-query
+minimum for any integer power alphabet; it does not supply a priced non-dyadic
+lowering. `research/dyadic-query-runs` batches several dyadic pairs and does not
+admit kernel 3. No competing open PR on this freestanding package was found in
+the refreshed overlap scan.
+
+## Open next questions
+
+1. Does a structured encoder make the odd-kernel shear usable inside an
+   admitted BIT supplier on moduli coprime to 3? First write one exact
+   source-return boundary and charge routing, precision and dispatch.
+2. What is the excluded-prime pattern for a general odd kernel `q`, and does
+   any small `q` improve a concrete paid profile after those exclusions?
+
+## Credit
+
+Prepared by Wentao Li ([VictorLiwentao](https://github.com/VictorLiwentao),
+[@Is__Victor](https://x.com/Is__Victor)) with substantial AI assistance, under
+Apache-2.0. Credit PR840/PR521/PR677 and the retained OpenAI tape framework.
+No independent human review is claimed. No new kappa.
diff --git a/research/triadic-query-pair/SOURCES.json b/research/triadic-query-pair/SOURCES.json
@@ -0,0 +1,36 @@
+{
+  "schema_version": 1,
+  "base_commit": "0b1cccf86e2f5132ed286a26ef0a681a495740ea",
+  "files": [
+    {
+      "path": "research/dyadic-query-pair/PROOF.md",
+      "usage": "input",
+      "git_blob": "aaba3e501de1f761c783ec7f0dc9307ef45ab5d7",
+      "sha256": "9a5ce93b866371189c797fbe8837c6a0bd22c5cf71ed84d60b9ae1c22a5ec83a"
+    },
+    {
+      "path": "research/dyadic-query-pair/README.md",
+      "usage": "input",
+      "git_blob": "3988b2cb22f81be52ca00500e1576d9f52206dab",
+      "sha256": "9b37c24a1056fa495f12e322ba56f592494f6f34516b0112be0db3f5cd6883d4"
+    },
+    {
+      "path": "research/dyadic-query-pair/CERTIFICATE.json",
+      "usage": "input",
+      "git_blob": "d201254cdd56e90471869caadaba5bc5eda8436c",
+      "sha256": "8267e9a468eca814b3b8641aae07ff71acadb303fd434ce3ef1984a258c69307"
+    },
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "ae6a99f28aa468649eb66a361e764d769363bf11",
+      "sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "06c96f5579bb67ed1b86d334e78fbed7da6945f1",
+      "sha256": "6686ad58f1fbdbc87134a78901b60fd53a20d3984355e91defbbebc9e9fa8075"
+    }
+  ]
+}